Question

let

$f(x)=7x^{1/5}(x^{3}-2)$.

evaluate the following specific values of $f$:
(a) $f(3)=$ 246.13
(b) $f(6)=$ 1153.48

Explanation:

Step1: Expand the function

$$\begin{align*} f(x)&=7x^{1/5}(x^{3}-2)\\ &=7x^{1/5 + 3}-14x^{1/5}\\ &=7x^{16/5}-14x^{1/5} \end{align*}$$

Step2: Differentiate using power rule

The power rule is \((x^n)^\prime=nx^{n - 1}\).
For \(y = 7x^{16/5}-14x^{1/5}\), then \(f^\prime(x)=7\times\frac{16}{5}x^{\frac{16}{5}-1}-14\times\frac{1}{5}x^{\frac{1}{5}-1}\)
\(f^\prime(x)=\frac{112}{5}x^{\frac{11}{5}}-\frac{14}{5}x^{-\frac{4}{5}}\)

Step3: Calculate \(f^\prime(3)\)

Substitute \(x = 3\) into \(f^\prime(x)\):

$$
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$$

Answer:

\(f^\prime(3)\approx249.90\)